Constant structure - Revision history

Hkm at 21:40, 7 August 2025

2025-08-07T21:40:44Z

← Older revision		Revision as of 21:40, 7 August 2025
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	A [[scale]] is said to be a '''constant structure''' ('''CS''') if its [[interval class]]es are distinct. That is, each [[interval size]] that occurs in the scale always ~~[[subtend]]s~~ the same number of scale steps. This means that you never get something like an interval being counted as a fourth one place, and a fifth another place.		A [[scale]] is said to be a '''constant structure''' ('''CS''') if its [[interval class]]es are distinct. That is, each [[interval size]] that occurs in the scale always spans the same number of scale steps. This means that you never get something like an interval being counted as a fourth one place, and a fifth another place.

	If a scale is a constant structure, that scale can be mapped to an [[isomorphic keyboard]] or similar isomorphic instrument such that each chord with the same interval structure can be played using the same fingering shape.		If a scale is a constant structure, that scale can be mapped to an [[isomorphic keyboard]] or similar isomorphic instrument such that each chord with the same interval structure can be played using the same fingering shape.

Fredg999: Text replacement - "Scale properties simplified" to "Glossary of scale properties"

2025-04-18T02:14:04Z

Text replacement - "Scale properties simplified" to "Glossary of scale properties"

← Older revision		Revision as of 02:14, 18 April 2025
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	* [[Gallery of CS Scales]]		* [[Gallery of CS Scales]]
	* [[~~Scale~~ properties ~~simplified~~]]		* [[Glossary of scale properties]]
	* [[epimorphic]]		* [[epimorphic]]
	* [http://tonalsoft.com/enc/c/constant-structure.aspx Constant structure] (Tonalsoft Encyclopedia)		* [http://tonalsoft.com/enc/c/constant-structure.aspx Constant structure] (Tonalsoft Encyclopedia)

Inthar: /* Related terminology */

2025-01-03T15:10:58Z

Related terminology

← Older revision		Revision as of 15:10, 3 January 2025
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	== ~~Related~~ terminology ==		== Novel terminology ==
	An interval that occurs in a scale is ''CS-consistent''{{idiosyncratic}} if it always subtends the same number of scale steps. A scale is thus CS if and only if all its intervals are CS-consistent. This term could be useful because someone might only care about certain primes in a subgroup being CS-consistent.		An interval that occurs in a scale is ''CS-consistent''{{idiosyncratic}} if it always subtends the same number of scale steps. A scale is thus CS if and only if all its intervals are CS-consistent. This term could be useful because someone might only care about certain primes in a subgroup being CS-consistent.

Inthar: /* Related terminology */

2025-01-03T15:03:37Z

Related terminology

← Older revision		Revision as of 15:03, 3 January 2025
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	== Related terminology ==		== Related terminology ==
	An interval that occurs in a scale is '''CS-consistent'''{{idiosyncratic}} if it always subtends the same number of scale steps. A scale is thus CS if and only if all its intervals are CS-consistent. This term could be useful because someone might only care about certain primes in a subgroup being CS-consistent.		An interval that occurs in a scale is ''CS-consistent''{{idiosyncratic}} if it always subtends the same number of scale steps. A scale is thus CS if and only if all its intervals are CS-consistent. This term could be useful because someone might only care about certain primes in a subgroup being CS-consistent.

	== See also ==		== See also ==

Inthar: /* Related terminology */

2025-01-03T15:02:25Z

Related terminology

← Older revision		Revision as of 15:02, 3 January 2025
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	== Related terminology ==		== Related terminology ==
	An interval that occurs in a scale is '''CS-consistent'''{{idiosyncratic}} if it always subtends the same number of scale steps. A scale is thus CS if and only if all its intervals are CS-consistent. This term could be useful because someone might only care about certain primes in a subgroup being ~~constantly mapped~~.		An interval that occurs in a scale is '''CS-consistent'''{{idiosyncratic}} if it always subtends the same number of scale steps. A scale is thus CS if and only if all its intervals are CS-consistent. This term could be useful because someone might only care about certain primes in a subgroup being CS-consistent.

	== See also ==		== See also ==

Inthar at 15:02, 3 January 2025

2025-01-03T15:02:02Z

← Older revision		Revision as of 15:02, 3 January 2025
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	\|}		\|}

			== Related terminology ==
			An interval that occurs in a scale is '''CS-consistent'''{{idiosyncratic}} if it always subtends the same number of scale steps. A scale is thus CS if and only if all its intervals are CS-consistent. This term could be useful because someone might only care about certain primes in a subgroup being constantly mapped.

	== See also ==		== See also ==

Bcmills: standardize on “is a constant structure” rather than “has constant structure”

2024-05-06T13:26:15Z

standardize on “is a constant structure” rather than “has constant structure”

← Older revision		Revision as of 13:26, 6 May 2024
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	A [[scale]] is said to ~~have~~ '''constant structure''' ('''CS''') if its [[interval class]]es are distinct. That is, each [[interval size]] that occurs in the scale always [[subtend]]s the same number of scale steps. This means that you never get something like an interval being counted as a fourth one place, and a fifth another place.		A [[scale]] is said to be a '''constant structure''' ('''CS''') if its [[interval class]]es are distinct. That is, each [[interval size]] that occurs in the scale always [[subtend]]s the same number of scale steps. This means that you never get something like an interval being counted as a fourth one place, and a fifth another place.

	If a scale ~~has~~ constant structure, that scale can be mapped to an [[isomorphic keyboard]] or similar isomorphic instrument such that each chord with the same interval structure can be played using the same fingering shape.		If a scale is a constant structure, that scale can be mapped to an [[isomorphic keyboard]] or similar isomorphic instrument such that each chord with the same interval structure can be played using the same fingering shape.

	The term "constant structure" was coined by [[Erv Wilson]]. In academic music theory, constant structure is called the partitioning property, but Erv got there first.		The term "constant structure" was coined by [[Erv Wilson]]. In academic music theory, constant structure is called the partitioning property, but Erv got there first.

	In terms of [[Rothenberg propriety]], strictly proper scales ~~have CS~~, and proper but not strictly proper scales do not. Improper scales generally do. However the [[22edo]] scale C D E vF# G ^Ab B C (<code>4-4-3-2-2-6-1</code>) has both ambiguity (C-vF# 4th equals vF#-C 5th) and contradiction (^Ab-B 2nd exceeds E-G 3rd). The contradiction makes it improper and the ambiguity makes it not CS.		In terms of [[Rothenberg propriety]], strictly proper scales are constant structures, and proper but not strictly proper scales are not. Improper scales generally are. However, the [[22edo]] scale C D E vF# G ^Ab B C (<code>4-4-3-2-2-6-1</code>) has both ambiguity (C-vF# 4th equals vF#-C 5th) and contradiction (^Ab-B 2nd exceeds E-G 3rd). The contradiction makes it improper and the ambiguity makes it not a CS.

	To determine if a scale ~~has~~ CS, all possible intervals between scale steps must be evaluated. An easy way to do this is with an [[interval matrix]], in which each entry gives the interval spanning the number of scale steps indicated by the column, beginning with step indicated by the row. In a CS scale, each interval in the matrix must appear in only one column, corresponding to the “constant” number of steps for that interval.		To determine if a scale is a CS, all possible intervals between scale steps must be evaluated. An easy way to do this is with an [[interval matrix]], in which each entry gives the interval spanning the number of scale steps indicated by the column, beginning with step indicated by the row. In a CS scale, each interval in the matrix must appear in only one column, corresponding to the “constant” number of steps for that interval.

	== Examples ==		== Examples ==
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	\| 2/1		\| 2/1
	\|}		\|}
	Note the highlighted intervals that occur in more than one column. For example, 5/4 may occur as either two or three steps of the scale. Thus, this scale ~~does~~ not ~~have~~ constant structure.		Note the highlighted intervals that occur in more than one column. For example, 5/4 may occur as either two or three steps of the scale. Thus, this scale is not a constant structure.

	=== Diatonic scales ===		=== Diatonic scales ===

Bcmills: diatonic example: removed table of note names

2024-05-04T03:10:42Z

diatonic example: removed table of note names

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 |}
-The corresponding note names:
+The highlighted intervals, from F to B and from B to F, are the same size in 12edo: 6\12, or 600 cents. From F to B, this interval spans four steps of our diatonic scale (an “augmented fourth”); but from B to F it spans five (a “diminished fifth”). Since the same interval spans different numbers of scale steps at different points in the scale, this scale is not a constant structure.
 However, in other tunings of the diatonic scale, the F–B and B–F intervals may have distinct sizes. For example, [[31edo]] (meantone) tunes F–B and B–F to 15\31 (581¢) and 16\31 (619¢) respectively:

Bcmills: note the practical significance of CS for isomorphic instruments

2024-05-03T15:17:06Z

note the practical significance of CS for isomorphic instruments

← Older revision		Revision as of 15:17, 3 May 2024
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	A [[scale]] is said to have '''constant structure''' ('''CS''') if its [[interval class]]es are distinct. That is, each [[interval size]] that occurs in the scale always [[subtend]]s the same number of scale steps. This means that you never get something like an interval being counted as a fourth one place, and a fifth another place.		A [[scale]] is said to have '''constant structure''' ('''CS''') if its [[interval class]]es are distinct. That is, each [[interval size]] that occurs in the scale always [[subtend]]s the same number of scale steps. This means that you never get something like an interval being counted as a fourth one place, and a fifth another place.

			If a scale has constant structure, that scale can be mapped to an [[isomorphic keyboard]] or similar isomorphic instrument such that each chord with the same interval structure can be played using the same fingering shape.

	The term "constant structure" was coined by [[Erv Wilson]]. In academic music theory, constant structure is called the partitioning property, but Erv got there first.		The term "constant structure" was coined by [[Erv Wilson]]. In academic music theory, constant structure is called the partitioning property, but Erv got there first.

Bcmills: clarified and expanded diatonic scale example and interval matrix discussion

2024-05-03T15:03:46Z

clarified and expanded diatonic scale example and interval matrix discussion

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